View publication

We study the fundamental question of how to define and measure the distance from calibration for probabilistic predictors. While the notion of perfect calibration is well-understood, there is no consensus on how to quantify the distance from perfect calibration. Numerous calibration measures have been proposed in the literature, but it is unclear how they compare to each other, and many popular measures such as Expected Calibration Error (ECE) fail to satisfy basic properties like continuity. We present a rigorous framework for analyzing calibration measures, inspired by the literature on property testing. We propose a ground-truth notion of distance from calibration: the 1\ell_1 distance to the nearest perfectly calibrated predictor. We define a consistent calibration measure as one that is a polynomial factor approximation to the this distance. Applying our framework, we identify three calibration measures that are consistent and can be estimated efficiently: smooth calibration, interval calibration, and Laplace kernel calibration. The former two give quadratic approximations to the ground truth distance, which we show is information-theoretically optimal. Our work thus establishes fundamental lower and upper bounds on measuring distance to calibration, and also provides theoretical justification for preferring certain metrics (like Laplace kernel calibration) in practice.

Related readings and updates.

When Does Optimizing a Proper Loss Yield Calibration?

Optimizing proper loss functions is popularly believed to yield predictors with good calibration properties; the intuition being that for such losses, the global optimum is to predict the ground-truth probabilities, which is indeed calibrated. However, typical machine learning models are trained to approximately minimize loss over restricted families of predictors, that are unlikely to contain the ground truth. Under what circumstances does…
See paper details

The Calibration Generalization Gap

This paper was accepted at the Workshop on Distribution-Free Uncertainty Quantification at ICML 2022. Calibration is a fundamental property of a good predictive model: it requires that the model predicts correctly in proportion to its confidence. Modern neural networks, however, provide no strong guarantees on their calibration— and can be either poorly calibrated or well-calibrated depending on the setting. It is currently unclear which factors…
See paper details